On a class of second order partial differential operators of non- principal type. (English) Zbl 0595.35122
Let P(x,D) be a second order linear differential operator with smooth coefficients and real value principal symbol \(p_ 2(x,\xi)\). Let \(q(x,\xi)=p_ 1(x,\xi)+(i/2)\sum p^{(j)}_{2(j)} (x,\xi)\) be its subprincipal symbol. Suppose that from \(p_ 2(x,\xi)=0\), \(d_{\xi}p_ 2(x,\xi)=0\) it follows that Im\(q(x,\xi)\neq 0\). Then at least one of the following estimates: \(\| u\|_ 1\leq C\| Pu\|_ 0\), \(\| u\|_ 1\leq C\| P^*u\|_ 0\) for \(u\in C_ 0^{\infty}(U)\) is valid.
Reviewer: Yu.V.Egorov
MSC:
| 35S99 | Pseudodifferential operators and other generalizations of partial differential operators |
| 35G05 | Linear higher-order PDEs |
References:
| [1] | Duistermaat, J. J. and Hörmander L., Fourier Integral Operators II,Acta Math., 128 (1972), 183–269. · Zbl 0232.47055 · doi:10.1007/BF02392165 |
| [2] | Gilioli, A. and Treves, F., An Example in the Solvability Theory of Linear PDE’s,Amer. J. Math., 96 (1974), 367–385. · Zbl 0308.35022 · doi:10.2307/2373639 |
| [3] | Mcnikoff, A., Pseudo-differential Operators with Double Characteristics,Math. Ann., 231 (1977), 145–180. · Zbl 0364.35050 · doi:10.1007/BF01361139 |
| [4] | Yamasaki A., On a Necessary Condition for the Local Solvability of Pseudo-differential Operators with Double Characteristics,Comm. in PDE’s, 5 (1980), 209–224. · Zbl 0436.35080 · doi:10.1080/0360530800882138 |
| [5] | Hörmander L., Differential Operators of Principal Type,Math. Ann., 140 (1960), 124–146. · Zbl 0090.08101 · doi:10.1007/BF01360085 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
